Years ago I carried a full SLR body, lenses, filters, tripod, etc. (plus film) for a multi-country trip through Asia.
My wife and I went with a friend from college and his wife. He carried an all-in-one auto-focus point and shoot Canon that was just larger than pocket size.
My pictures came back and they were wonderful. His pictures came back and they were wonderful too. Mine weren't nearly enough better than his to justify packing the weight.
Since then, I've gone with the smallest point and shoot I can find. A neck strap from a thumb drive is a perfect replacement for the wrist strap they come with.
The features I like: Quick shutter release, large viewing window, at least 5x optical zoom, battery life (hopefully better than 200 shots on a charge), small, and good optics for good pictures. Most also come with a host of other options for different shooting situations.
Right now I have a pocket size Nikon with 7x optical zoom. It was about $200 at Costco.
I would avoid brands like Kodak, Panasonic, Casio, etc. Ultimately, the lens quality limits picture quality more than anything else - only the brands with great lenses will deliver great pictures. I always get great results from Canon or Nikon. Others are talking up Sony's new optics too, but I've never owned one.
I work with many professional photographers who suggest a Canon G-10. It's larger than pocket size but a lot smaller than an SLR and lens. Apparently most pros carry a G-10 as a backup camera, they run about $500 (one of the features they really like, that I wouldn't use, is the hot shoe for adding an external flash).